What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing convolution?
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The "endearing" behaviour this OP exhibited on this page (whose traces are mainly erased now) continued on this other page. – Did Sep 03 '16 at 07:38
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The meanings of convolution and cross-correlation are quite different.
To see what happens in a simple setting, consider some independent integer valued random variables $X$ and $Y$ with respective distributions $p=(p_n)_n$ and $q=(q_n)_n$. Then:
The convolution $p\ast q$ is the distribution $s=(s_n)_n$ defined by $$s_n=\sum\limits_kp_kq_{n-k}=P[X+Y=n]$$ for every $n$. Thus, $p\ast q$ is the distribution of $X+Y$.
The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by $$c_n=\sum\limits_kp_kq_{n+k}=P[Y-X=n]$$ for every $n$. Thus, $p\circ q$ is the distribution of $Y-X$.
To sum up, considered at the level of distributions of independent random variables, the composition $\ast$ acts as an addition while the composition $\circ$ acts as a difference.
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3I don't understand this. I'm (i) confused on the notation (p_n)_n (what do both subscript n do here), and (ii) what is big P - probability? Why does X+Y = n? Sorry for being dumb! – friartuck Apr 24 '18 at 05:01
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2@Prof Re (i), $(p_n)n$ is the archi classical notation for a sequence $(\ldots,p{-2},p_{-1},p_0,p_1,p_2,\ldots)$ as opposed to a specific value $p_n$ for a specific index $n$. Re (ii) and (iii), sorry but your queries make little sense: "what is big P - probability?" ??? Well... yes, big-P denotes the probability (what else?). "Why does X+Y = n?" Does it? Recall that $X+Y$ is a random variable hence $A={X+Y=n}$ is an event and $P[X+Y=n]$ stands for $P(A)$. Never saw the string of characters:
P[X=x]even once in your life before? – Did Apr 24 '18 at 05:54